For any two complex numbers z₁ and z₂, prove that Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

Solution:

Let us assume that Z₁ = x₁ + iy₁ and Z₂ = x₂ + iy₂ are two complex numbers.

Then Re Z₁ = x₁, Im Z₁ = y₁;  and Re Z₂ = x₂. Im Z₂ = y₂ ... (1)

LHS = Re (z₁z₂)

= Re ( (x₁ + iy₁) (x₂ + iy₂) )

= Re (x₁y₁ + ix₁y₂ + ix₂ y₁ - y₁ y₂)

= x₁y₁ - y₁ y₂

= Re z₁ Re z₂ – Im z₁ Im z₂ (From (1))

Hence proved that Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

NCERT Solutions Class 11 Maths Chapter 5 Exercise ME Question 2

For any two real complex z₁ and z₂, prove that Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂

Summary:

Hence proved that Re (z₁z₂) = Re z₁ Re z₂ – Im z₁ Im z₂